Simplify the following expression and state the condition under which the simplification is valid. You can assume that $p \neq 0$. $q = \dfrac{8p + 8}{-10} \div \dfrac{5p^2 + 5p}{9} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $q = \dfrac{8p + 8}{-10} \times \dfrac{9}{5p^2 + 5p} $ When multiplying fractions, we multiply the numerators and the denominators. $q = \dfrac{ (8p + 8) \times 9 } { -10 \times (5p^2 + 5p) } $ $ q = \dfrac {9 \times 8(p + 1)} {-10 \times 5p(p + 1)} $ $ q = \dfrac{72(p + 1)}{-50p(p + 1)} $ We can cancel the $p + 1$ so long as $p + 1 \neq 0$ Therefore $p \neq -1$ $q = \dfrac{72 \cancel{(p + 1})}{-50p \cancel{(p + 1)}} = -\dfrac{72}{50p} = -\dfrac{36}{25p} $